Ultra-precision freeform surfaces are important to the development of complex and micro-optical-electro-mechanical devices used in many photonics and telecommunication products such as F-theta lenses for laser printers. However, these surfaces are complex and large scale surface topologies with shapes that generally possesses no rotational symmetry. The geometry of these surfaces usually cannot be generalized by a single optical equation such as the universal optics equation used for aspheric surfaces. To fulfill the stringent requirements for advanced optics applications, these ultra-precision freeform surfaces are usually fabricated by ultra-precision freeform machining technology, and the surface finishes with sub-micrometer form accuracy requirement in at least sub-micrometer, often nanometer, range.
Due to the geometrical complexities of these ultra-precision freeform surfaces, it is difficult to characterize the form accuracy and surface quality of freeform optical surfaces. Although some surface parameters have been proposed by the ISO project, their applicability of practical inspection is limited to optical surfaces possessing rotational symmetry. The general principle of form characterization of the freeform surfaces was analyzed by Xiong [Y. L. Xiong, The mathematic approach of precision measurement, Beijing: Metrology Press, China, 1989]. He thought the key was to make the measured surface composed of a series of discrete points overlapping with the theoretically designed surface in the conditions of the least area with the shift and rotation of coordinates. By making the sum of distance of all measured points from the theoretic designed surface, the form error was obtained. The characterization results are usually obtained only when the effect of systematic errors was eliminated.
However, two non-linear problems must be solved. One of them is the solution of the coordinate transfer to ensure the consistence of the measured coordinates with the designed coordinates while the other is to compute the projection of the measured points after the coordinate transfer on the theoretic surface. The best matching method for form characterization of the freeform surfaces was proposed [A. H. Rentoul and G. Medland, Interpretation of errors from inspection results, Computer Integrated Manufacturing Systems, 1994, 7(3): 173-178]. By shifting and rotating the coordinates, the feature points, feature lines and feature planes of the measured and design surfaces were overlapped completely and the best alignment was attained. The systematic error from the coordinate disagreement was removed. The method was based not only on the least square criterion but also the least area principle. Based on the best matching method, Wang et al. [P. J. Wang, J. H. Chen, and Z. Q. Li, A new algorithm for the profile Error of a parameter surface, Journal of Huazhong University of Science & Technology, 1997, 25(3): 1-4] presented a new algorithm with iterative approximation of the form error of freeform surfaces. After the matching of the coordinates, the least area of the approximated form error was obtained by the iterative coordinate transfer.
Kase et al. [K. Kase, A. Makinouchi, T. Nakagawa, et al., Shape error evaluation method of free-form surfaces, Computer-aided design, 1999, 31: 495-505] used the main curvature change of the measured and design surface to conduct local characterization while the global characterization was conducted based on the normal vector. Hua et al. [H. Hua, Y. B. Li, K. Cheng, et al., A practical evaluation approach towards from deviation for two-dimensional contours based on coordinate measurement data, International Journal of Machine Tools and Manufacture, 2000, 40(1): 259-275] divided the 2D contour profiles into straight and curved parts to characterize the form error. Each curve was represented by its optimal interpolation circular arc segment. The form deviation of a measured point is defined as the distance from the point to its corresponding straight or circular segment. The deviation of the geometrical form of the contour profile is evaluated using the least square method. Based on the minimum principle, Yu et al. [Y. Yu, J. Lu and X. C. Wang, Modeling and analysis of the best matching free-form surface measuring, Mechanical Science and Technology, 2001, 20(3): 469-471] used the rotation and movement of the freeform surface to perform the pre-location of measured surface. The systematic error was eliminated effectively by the best matching. According to an improved algorithm of coordinate alternation, the precise adjustment of measured coordinates was accomplished and the profile error of the freeform surface was obtained.
Although the significant achievement has been obtained in the characterization of freeform surface, it was found that the above methods are inadequate to characterize ultra-precision freeform surfaces with form error in sub-micrometer range. There is a need to develop new method to characterize surfaces, particularly freeform surfaces, so that sub-micrometer form error can be identified.